In reply to mypyrex:
I have one of the Peak Design clips, and while it doesn't satisfy your requirements for well-protected, I've used it pretty much everywhere from scrambling in North Wales to multipitch rock and alpine routes without any problems. An inconvenience for sure, but carrying a camera of that size and having it to hand is always going to be a compromise. I've accepted that the body is going to get scratched up, and the system otherwise works well.
https://www.peakdesign.com/store/?gclid=CN3A-O6OwMYCFQIcwwodb80FOg
Definitely use some sort of lanyard though to avoid dropping the camera. The Peak Design leash works well, but even better I've recently discovered is a Black Diamond spinner leash. Almost like it was designed for the purpose, and better than any camera-specific product I've found so far!